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Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","8"],["4","32","64"]],"card_rows":[["1"],["2","8"],["4","32","64"],["8","96","384","512"]],"row_sums_preview":["1","10","100","1000","10000","100000","1000000","10000000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013611"],"edges":["e-d7d2422771947afc"],"text":"A130123 \u2192 basis A013611"},"headline":"A130123 \u2192 basis A013611","chain":null}},{"id":"l-d8026226b4b5457a","kind":"fingerprint_related","node_a_id":"n-ef5c9478e7d83dab","node_b_id":"n-6e0864853f49fb26","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-ef5c9478e7d83dab","other":{"id":"n-ef5c9478e7d83dab","name":"derivative of Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["8","2"]],"card_rows":[["1"],["8","2"],["48","24","3"],["256","192","48","4"]],"row_sums_preview":["0","1","10","75","500","3125","18750","109375"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3dcaebc50e901d48","steps":["D"],"edges":["e-62aaaa59f91d42c1"],"text":"A038231 \u2192 D"},"headline":"A038231 \u2192 D","chain":null}},{"id":"l-ecc4ea45e06a5a79","kind":"fingerprint_related","node_a_id":"n-6e0864853f49fb26","node_b_id":"n-872d1ad8224c1da6","fingerprint":"f6-03d7b8417365c509","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-03d7b8417365c509"},"other_id":"n-872d1ad8224c1da6","other":{"id":"n-872d1ad8224c1da6","name":"basis_map_A013611 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-03d7b8417365c509","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","4"],["2","16","32"]],"card_rows":[["1"],["1","4"],["2","16","32"],["6","72","288","384"]],"row_sums_preview":["1","5","50","750","15000","375000","11250000","393750000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013611"],"edges":["e-d759043f254c4b9e"],"text":"A136572 \u2192 basis A013611"},"headline":"A136572 \u2192 basis A013611","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       1;\n   2:      -1,        -4;\n   3:       2,        16,         32;\n   4:      -6,       -72,       -288,        -384;\n   5:      24,       384,       2304,        6144,         6144;\n   6:    -120,     -2400,     -19200,      -76800,      -153600,      -122880;\n   7:     720,     17280,     172800,      921600,      2764800,      4423680,       2949120;\n   8:   -5040,   -141120,   -1693440,   -11289600,    -45158400,   -108380160,    -144506880,     -82575360;\n   9:   40320,   1290240,   18063360,   144506880,    722534400,   2312110080,    4624220160,    5284823040,    2642411520;\n  10: -362880, -13063680, -209018880, -1950842880, -11705057280, -46820229120, -124853944320, -214035333120, -214035333120, -95126814720;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n-1+(4*n-4)*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(n - 1 + (4*n - 4)*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = -(n - 1 + (4*n - 4)*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["basis A013611"],"edges":["e-e262ea1acc16711f"],"text":"basis-map coefficients of A021012 \u2192 basis A013611"},"headline":"basis-map coefficients of A021012 \u2192 basis A013611","chain":null}